Pineapple adds sweetness and a tropical twist to this classic cocktail. Make sure to choose a ripe one: It should be very fragrant, with no soft spots or bruises. Or use fresh precut pineapple.
Bring sugar and 1 cup water to a boil in a small saucepan, stirring to dissolve sugar. Remove from heat, add mint, and let cool completely.
Meanwhile, place pineapple in a blender and purée until smooth.
Strain mint syrup through a fine-mesh sieve into pineapple purée and pulse to combine. Transfer to a jar, cover, and chill.
For each cocktail, muddle 2 pieces of pineapple, 3 lime wedges, and 1 tablespoon mint leaves in a cocktail shaker until lightly crushed.
Add 2 ounces rum and 2 ounces pineapple-mint purée to cocktail shaker, fill with ice, cover, and shake until outside of shaker is frosty, about 30 seconds.
Pour drink into a highball glass (do not strain) and top off with club soda. Garnish with a lime wedge and serve with a straw.
DO AHEAD: Pineapple-mint purée can be made 3 days ahead. Keep chilled.
